Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 102, pp. 1–18.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, DOI: 10.58997/ejde.2025.102

Lq-REGULARITY ESTIMATES FOR DOUBLE OBSTACLE PROBLEMS

WITH QUASILINEAR OPERATORS AND SCHRÖDINGER-TYPE

LOWER ORDER TERMS

MIKYOUNG LEE, JIHOON OK

Abstract. We derive interior and boundary Lq-regularity estimates for double obstacle prob-

lems involving quasilinear operators of p-Laplacian type and lower-order terms with nonnegative

potential functions satisfying a reverse Hölder type condition. We prove that the Lq norms of
the gradient of a solution to the obstacle problem, as well as the lower-order term, can be esti-

mated by the Lq norms of the data and the gradients of the obstacles. Moreover, the relevant

constants in the Lq estimates depend only on the constant in the reverse Hölder type condition
for the potential and are independent of the potential.

1. Introduction

Obstacle problems have long played a central role in mathematical models where physical
constraints are present. A classical motivation arises from studying the equilibrium configuration
of a stretched membrane that is restricted from below by a fixed obstacle. Mathematically, this
leads to constrained minimization problems, where the goal is to find energy-minimizing functions
subject to a pointwise lower bound. Such formulations naturally belong to the framework of
variational inequalities and establish a strong connection with both the calculus of variations and
partial differential equations. These types of problems frequently emerge in real-world scenarios,
including fluid flow through porous materials, thermally constrained systems, elasto-plastic models
in mechanics, optimal control theory, and pricing models in finance.

While much progress has been made for single obstacle problems, many practical systems involve
situations where a solution is constrained between both lower and upper bounds. This gives rise to
the double obstacle problem, where the solution is restricted to lie between two obstacle functions.
These problems are not only natural generalizations of the single obstacle case but also present
new analytical challenges, especially in quasilinear settings and in the presence of lower-order
terms, which can significantly affect the behavior and regularity of solutions. Motivated by these
challenges, we aim to contribute to the theory by analyzing regularity properties of solutions to
double obstacle problems governed by quasilinear operators with additional lower-order terms.

In the calculus of variations, the double obstacle problem is to find a function (solution) that
minimizes the energy ∫

Ω

f(x, v,Dv) dx

among the functions v ∈ W 1,1(Ω) with the same boundary value, such that ψ1 ≤ v ≤ ψ2 for
some obstacle functions ψ1 and ψ2 on Ω ⊂ Rn. Note that the single obstacle problem is the case
ψ2 = ∞. If we assume a suitable p-growth condition with 1 < p <∞ on f , then the solution u of
the double obstacle problem is in W 1,p(Ω) and satisfies the variational inequality∫

Ω

Dξf(x, u,Du) ·D(u− φ) + ∂vf(x, u,Du)(u− φ) dx ≤ 0

2020 Mathematics Subject Classification. 35J87, 35J92, 35J10, 35B65.
Key words and phrases. Obstacle problem; Schrödinger operator; Lq-estimates; nonnegative potential;
p-Laplacian.
©2025. This work is licensed under a CC BY 4.0 license.
Submitted July 2, 2025. Published October 30, 2025.

1



2 M. LEE, J. OK EJDE-2025/102

for all φ ∈ W 1,p(Ω) such that ψ1 ≤ φ ≤ ψ2 and u − φ ∈ W 1,p
0 (Ω). Here, Dξf(x, v, ξ) is the

gradient of f with respect to ξ ∈ Rn, and fv(x, v, ξ) is the partial derivative of f with respect
to v ∈ R. A fundamental observation in the study of regularity theory for obstacle problems is
that the regularity of the solution is strongly governed, often even determined, by how smooth the
obstacle function is. Many researchers have explored the regularity properties of obstacle problems
with p-growth. Notably, for single obstacle problems, Michael and Ziemer [29] proved that if the
obstacle function is Hölder continuous, then the solution is also Hölder continuous. Later, Choe
[11] showed that the Hölder continuity of the gradient of the obstacle function ensures the same for
the gradient of the solution. Bögelein, Duzaar and Mingione [4] obtained Lq-regularity estimates
for the gradient of the solution. For further regularity results on single obstacle problems, we
refer to, for example, [3, 6, 12, 19, 26, 27, 33, 38]. On the other hand, for regularity results
concerning the double obstacle problems, we refer to [13] for pointwise regularity, [22, 37] for
Hölder continuity for the solution, [25, 31] for Hölder continuity for the gradient of the solution,
and [7, 8] for Lq-estimates for the gradient of the solution.

We study Lq-regularity theory on variational inequalities when the function f has the form

f(x, v, ξ) = g(x, ξ) +
1

s
V (x)|v|s − |F (x)|p−2F (x) · ξ, (x, v, ξ) ∈ Rn × R× Rn,

where 1 < p < ∞, 1 < s < p∗, F (x) is a given Rn-valued function, and V (x) is a nonnegative
function, often referred to as a potential function, satisfying certain integrability conditions. In
this setting, we have that Dξf(x, v, ξ) = Dξg(x, ξ)−|F (x)|p−2F (x), and fv(x, v, ξ) = V (x)|v|s−2v.
More generally, we consider a general Rn-vector valued function a(x, ξ) of the p-Laplacian type,
replacing Dξg(x, ξ). The corresponding quasilinear elliptic equation then takes the form

− div a(x,Du) + V |u|s−2u = − div(|F |p−2F ). (1.1)

In the simplest case when p = s = 2 and a(x, ξ) ≡ ξ, this reduces to

(−∆+ V )u = −∆u+ V u = −divF,

where −∆+ V is known as the Schrödinger operator. For this equation, Lq-regularity theory has
been studied by Shen [39, 40], and by Auscher and Ben [2]. Specifically, in [40], Shen proved that
if V ∈ Bγ for some γ > n/2 (see Section 2.1 for the definition of Bγ), then

∥Du∥L2q(Rn) + χ{q≤γ}∥V 1/2u∥L2q(Rn) ≤ c∥F∥Lq(Rn)

for all (γ∗)′/2 ≤ q ≤ γ∗/2. Here, χ{q≤γ} = 1 if q ≤ γ and χ{q≤γ} = 0 otherwise, and the constant
c > 0 depends only on n, γ, q, and the relevant constant of the Bγ condition, and is independent
of the function V . For regularity results on linear equations associated with the Schrödinger
operator, we refer to [5, 10, 14, 20, 21, 35, 47, 48, 50]. In recent years, Lq-regularity estimates
have been successfully established for the corresponding nonlinear equations with p-growth in the
form of (1.1) by the authors [23, 24]. We refer to [32, 42, 44, 45, 46, 49] for Lq-regularity theory
on nonlinear problems with Schrödinger-type lower order terms.

In this article, we derive Lq estimates for |Du|p and V |u|s, where u is a function satisfying
the given variational inequality arising from the double obstacle problem associated with (1.1).
Notably, the paper [32] considered double obstacle problems concerned with (1.1) in the case s = p,
obtaining Lq-regularity estimates. However, their estimates are not sharp, as the right-hand side
contains the additional terms involving V |ψ1|p and V |ψ2|p. In contrast, our estimates exclude
these terms, thereby providing sharper bounds and closely matching the estimates obtained in
[24]. We also note that Lq estimates for the single obstacle problem associated with (1.1) have
not been studied; our results can be readily adapted to this simpler case.

Our proof is based on a comparison scheme that reduces the analysis to auxiliary problems
for which sharper estimates are available. The key idea is to construct three carefully designed
comparison problems: a variational inequality associated with a single obstacle problem, a non-
homogeneous equation in which the forcing term contains the super obstacle function, and the
corresponding homogeneous equation. We emphasize that all these problems include the lower-
order terms, which ultimately contribute to deriving the desired sharp estimates. A crucial step
in this scheme involves comparing the solution of a localized double obstacle problem with that



EJDE-2025/102 DOUBLE OBSTACLE PROBLEMS 3

of a suitably chosen single obstacle problem. This comparison effectively reduces the two-sided
constraint to a one-sided one, allowing us to circumvent the inherent difficulties of the double
obstacle setting. Once this reduction is achieved, the remainder of the argument proceeds in
close analogy with the analysis of the single obstacle case. In particular, we derive comparison
estimates for the gradient of the solution and the lower-order terms separately. This decoupling
enables us to handle them independently in the final stage of establishing the Lq estimates, via
the approach introduced by Mingione in [1, 30]. At this stage, we also make essential use of the
higher integrability results for homogeneous equations established in [24] (see Section 2.2), which
are specifically tailored to nonlinear elliptic equations with Schrödinger-type lower order terms.

1.1. Main results. Throughout the paper, let Ω be a bounded domain in Rn with n ≥ 2, 1 <
p <∞, and 1 < s < p∗ (see (2.1) for the definition of p∗). We always assume that F ∈ Lp(Ω,Rn),
and ψ1, ψ2 ∈ W 1,p(Ω) are obstacle functions satisfying that ψ1 ≤ 0 ≤ ψ2 a.e. in Ω. Note that
the condition for the obstacles is stronger than ψ1 ≤ ψ2 a.e. in Ω and max{ψ1, 0},max{−ψ2, 0} ∈
W 1,p

0 (Ω), which are typically assumed in zero Dirichlet boundary value problems. This stronger
condition is crucial for the comparison estimates to handle the lower-order term V |u|s−2u, see
Section 3. Then we consider the admissible set

A0(Ω) = {φ ∈W 1,p
0 (Ω) : ψ1 ≤ φ ≤ ψ2 a.e. in Ω}.

The potential V : Rn → [0,∞) is assumed to satisfy that V ∈ Lγ0

loc(Rn), where

γ0 :=


np

np−s(n−p) , when 1 < p < n,

any number larger than 1, when p = n,

1, when p > n.

(1.2)

and to be taken in an appropriate class Bγ (see (2.2)). Note that γ0 = n
p if s = p ∈ (1, n). The

nonlinearity a : Rn × Rn → Rn, which is a vector-valued Carathéodory function (i.e., a(x, ξ) is
measurable in the x-variable and continuous in the ξ-variable), is assumed to satisfy that a(x, ·) ∈
C1(Rn \ {0},Rn) for each x ∈ Ω, and the growth and ellipticity conditions:

|a(x, ξ)|+ |Dξa(x, ξ)||ξ| ≤ L|ξ|p−1, (1.3)

Dξa(x, ξ) η · η ≥ ν|η|2|ξ|p−2 (1.4)

for a.e. x ∈ Rn and any ξ, η ∈ Rn (ξ ̸= 0) and for some constants 0 < ν ≤ L. The prototype of
the nonlinearity a is

a(x, ξ) = (A(x)ξ · ξ)
p−2
2 A(x)ξ, (1.5)

where A : Rn → Rn2

is an n× n matrix satisfying that

ν|η|2 ≤ A(x)η · η and |A(x)| ≤ L, x, η ∈ Rn,

for some 0 < ν ≤ L. We remark that condition (1.4) implies the monotonicity condition

(a(x, ξ)− a(x, η)) · (ξ − η) ≥ ν̃
(
|ξ|2 + |η|2

) p−2
2 |ξ − η|2 (1.6)

for some ν̃ > 0 depending on p and ν, for any ξ, η ∈ Rn and a.e. x ∈ Rn.
We deal with the function u ∈ A0(Ω) that satisfies the variational inequality∫

Ω

a(x,Du) ·D(u− φ) dx+

∫
Ω

V |u|s−2u(u− φ) dx ≤
∫
Ω

|F |p−2F ·D(u− φ) dx (1.7)

for all φ ∈ A0(Ω). Notice that under the above setting, the existence and uniqueness of a function
satisfying (1.7), or the variational inequalities appearing in this paper, are ensured by general
existence theory for monotone operators; see, for instance, [16, 41].

Now we present the main results of this paper. Definitions of additional conditions on the
potential V , the nonlinearity a and the domain Ω, which are necessary for obtaining Lq estimates,
will be introduced in the next section. The first result provides local Lq estimates in both the
interior and boundary regions. For the definitions of p∗, γ∗ and p̃, see (2.1).



4 M. LEE, J. OK EJDE-2025/102

Theorem 1.1. Let 1 < p <∞ and 1 < s < p∗, and let a : Rn ×Rn → Rn satisfy (1.3) and (1.4),
F ∈ Lp(Ω,Rn), ψ1, ψ2 ∈ W 1,p(Ω) be such that ψ1 ≤ 0 ≤ ψ2 a.e. in Ω, and V : Rn → [0,∞) be
such that V ∈ Lγ0

loc(Rn) with γ0 given in (1.2). For each p, γ and q satisfying specific conditions
given below, there exists a small δ = δ(n, p, q, L, ν, γ) > 0 such that if a is (δ,R)-vanishing, Ω is a
(δ,R)-Reifenberg flat domain for some R > 0, and u ∈ A0(Ω) satisfies the variational inequality
(1.7) with V ∈ Bγ , then the following estimates hold for any x0 ∈ Ω and r ∈ (0, R2 ]: denote

Ψ := |F |p + |Dψ1|p + |Dψ2|p. (1.8)

(1) If 1 < p <∞, p̃ < γ <∞, and 1 < q < γ∗(p−1)
p ,

−
∫
Ωr(x0)

|Du|pq dx ≤ c
(
−
∫
Ω2r(x0)

|Du|p dx
)q

+ c−
∫
Ω2r(x0)

Ψq dx. (1.9)

(2) If p ≥ 2, 1 < γ <∞, and 1 < q < γ,

−
∫
Ωr(x0)

[V |u|s]q dx ≤ c
(
−
∫
Ω2r(x0)

V |u|s dx
)q

+ c−
∫
Ω2r(x0)

Ψq dx. (1.10)

(3) If 1 < p < 2, n
p ≤ γ <∞, and 1 < q < γ,

−
∫
Ωr(x0)

[V |u|s]q dx ≤ c
(
−
∫
Ω2r(x0)

|Du|p + V |u|s dx
)q

+ c−
∫
Ω2r(x0)

Ψq dx. (1.11)

Here, the constants c > 0 depend on n, p, L, ν, s, γ, bγ , and q.

For the upper bound γ of the exponent q, we refer to the comments in [24, Remarks 2.6 and
2.7]. The second result provides global Lq estimates. As a direct consequence of Theorem 1.1, we
obtain the following global estimates by employing a standard covering argument, as used in the
proof of [23, Corollary 2.6]. Therefore we omit the proof.

Theorem 1.2. Let 1 < p <∞ and 1 < s < p∗, and let a : Rn ×Rn → Rn satisfy (1.3) and (1.4),
F ∈ Lp(Ω,Rn), ψ1, ψ2 ∈ W 1,p(Ω) be such that ψ1 ≤ 0 ≤ ψ2 a.e. in Ω, and V : Rn → [0,∞) be
such that V ∈ Lγ0

loc(Rn) with γ0 given in (1.2). For each p, γ and q satisfying specific conditions
given below, there exists a small δ = δ(n, p, q, L, ν, γ) > 0 such that if a is (δ,R)-vanishing, Ω is a
(δ,R)-Reifenberg flat domain for some R > 0, and u ∈ A0(Ω) satisfies the variational inequality
(1.7) with V ∈ Bγ , then the following estimates hold: recall Ψ given in (1.8).

(1) If 1 < p <∞, p̃ < γ <∞ and 1 < q < γ∗(p−1)
p , then

∥Du∥Lpq(Ω) ≤ c
(diam(Ω)

R

)n(q−1)

∥Ψ∥Lq(Ω).

(2) If p ≥ 2, 1 < γ <∞ and 1 < q < γ, or if 1 < p < 2, n
p ≤ γ <∞ and 1 < q < γ, then

∥V 1/p|u|s/p∥Lpq(Ω) ≤ c
(diam(Ω)

R

)n(q−1)

∥Ψ∥Lq(Ω).

Here, the constants c > 0 depend on n, p, L, ν, s, γ, bγ , and q.

Remark 1.3. The nonnegative potential V is assumed to satisfy two conditions: V ∈ Lγ0

loc(Rn)
and V ∈ Bγ . The first condition guarantees the existence of a function u satisfying the variational
inequality (1.7). Note that if 1 < p < n, then p̃ < γ0 and hence γ may be smaller than γ0. In this
case, both conditions on V are required for the above results. On the other hand, if p ≥ n, or if
1 < p < n and γ ≥ γ0, then the second condition implies the first.

The remainder of the paper is organized as follows. In Section 2, we present notation, the main
assumptions and preliminary lemmas essential for our analysis. Section 3 is devoted to establishing
key comparison estimates that play a central role in our approach. In Section 4, we provide the
proof of our main theorem.



EJDE-2025/102 DOUBLE OBSTACLE PROBLEMS 5

2. Preliminaries

We begin with standard notation and definitions. For any y ∈ Rn and r > 0, we denote by Br(y)
the open ball in Rn centered at y with radius r. Given a domain Ω ⊂ Rn, we define the truncated
domains by Ωr(y) := Br(y) ∩ Ω and ∂wΩr(y) := Br(y) ∩ ∂Ω, which represent, respectively, the
portion of Ω and its boundary within the ball Br(y). For simplicity of notation, when the center
y is the origin or not relevant, we write Br := Br(y) and Ωr := Ωr(y). For a measurable function
f : U → R with U ⊂ Rn, we use the notation

(f)U := −
∫
U

f dx =
1

|U |

∫
U

f dx

to denote the average value of f over the set U , where |U | stands for the Lebesgue measure of
U . Furthermore, we define the positive part of f by f+ := max{f, 0}. For f ∈ W 1,p(Ωr(y)), the
boundary condition “f = 0 on ∂wΩr(y)” is understood in the sense that the extension of f by
zero to Br(y) belongs to W

1,p(Br(y)). In addition, we use the notation

p∗ :=

{
np
n−p , if 1 < p < n,

∞, if p ≥ n,
and p̃ :=

{
np

np−n+p , if 1 < p < n,

1, if p ≥ n.
(2.1)

We use the following standard iteration lemma, its proof can be found in [18].

Lemma 2.1. Let f : [a, b] → R be a bounded nonnegative function. Suppose that for any τ1, τ2
with 0 ≤ a ≤ τ1 < τ2 ≤ b,

f(τ1) ≤ τf(τ2) +
C1

(τ2 − τ1)β
+ C2

where C1, C2 ≥ 0, β > 0 and 0 ≤ τ < 1. Then we have

f(τ1) ≤ c
( C1

(τ2 − τ1)β
+ C2

)
for some constant c = c(β, τ) > 0.

2.1. Main assumptions. We consider a nonnegative function V : Rn → [0,∞), which serves as
a potential in the problem under consideration. We assume that V belongs to the class Bγ for
some γ > 1, denoted by V ∈ Bγ , meaning that V ∈ Lγ

loc(Rn) and there exists a constant bγ > 0
such that the following reverse Hölder inequality holds:(

−
∫
B

V γ dx
)1/γ

≤ bγ −
∫
B

V dx (2.2)

for every ball B ⊂ Rn. The class Bγ , which includes a broad range of functions such as all
nonnegative polynomials, was introduced independently by Muckenhoupt [28] in the study of
weighted norm inequalities and by Gehring [15] in the theory of quasiconformal mappings. A
representative example of a function in this class is V (x) = |x|−n/γ which actually belongs to the
Bp̃ class for every p̃ < γ. Moreover, the class Bγ is closely related to the Muckenhoupt class Ap.
We refer the reader to [17, Chapter 9] for a detailed account of the properties of the class Bγ and
its relationship with Muckenhoupt weights.

The following two definitions describe the structural assumptions imposed on the nonlinearity
a and the geometry of the domain Ω.

Definition 2.2. Let δ,R > 0, and a : Rn × Rn → Rn satisfy (1.3) and (1.4). We say that a is
(δ,R)-vanishing if

sup
0<ρ≤R

sup
y∈Rn

−
∫
Bρ(y)

|Θ
(
a, Bρ(y)

)
(x)| dx ≤ δ,

where

Θ (a, Bρ(y)) (x) := sup
ξ∈Rn\{0}

∣∣a(x, ξ)− (a(·, ξ))Bρ(y)

∣∣
|ξ|p−1

and

(a(·, ξ))Bρ(y) := −
∫
Bρ(y)

a(x, ξ) dx.



6 M. LEE, J. OK EJDE-2025/102

The condition in the above definition implies that the map x 7→ a(x, ξ)/|ξ|p−1 belongs to the
local bounded mean oscillation (BMO) space with the BMO semi-norm controlled by δ for all
ξ ∈ Rn. Consequently, the nonlinearity a is allowed to be discontinuous in the x-variable. In
particular, in the model case (1.5), the definition implies that A(·) is locally BMO.

Definition 2.3. Let δ ∈ (0, 1/8) and R > 0. A domain Ω ⊂ Rn is said to be a (δ,R)-Reifenberg flat
domain if for every x ∈ ∂Ω and every ρ ∈ (0, R], there exists a coordinate system {y1, y2, . . . , yn}
which may depend on ρ and x, such that in this coordinate system x = 0 and that

Bρ(0) ∩ {yn > δρ} ⊂ Bρ(0) ∩ Ω ⊂ Bρ(0) ∩ {yn > −δρ}.

The restriction δ < 1/8 in the above definition is standard in the literature and arises naturally
in connection with Sobolev embedding results (see, e.g., [43]). In our setting, this restriction is not
essential, as we will only consider sufficiently small values of δ. Lipschitz domains with Lipschitz
constants at most δ are special cases of (δ,R)-Reifenberg flat domains for some R > 0. Notably,
the Reifenberg class allows for more general geometries that are not necessarily graph-defined.
Furthermore, the (δ,R)-Reifenberg flat domain Ω satisfies the measure density condition

1 ≤ sup
0<ρ≤R

sup
y∈Ω

|Bρ(y)|
|Ωρ(y)|

≤
( 2

1− δ

)n

≤
(16
7

)n

. (2.3)

We refer to [9, 34, 36, 43] for further information on Reifenberg flat domains and their applications.

2.2. Estimates for homogeneous equations. We consider the homogeneous Dirichlet problem

−div a(x,Dh) + V |h|s−2h = 0 in Ω2r,

h = 0 on ∂wΩ2r, if B2r ̸⊂ Ω,
(2.4)

where Ω2r = Ω2r(x0) for some x0 ∈ Ω, a : Rn × Rn → Rn satisfies

|a(x, ξ)| ≤ L|ξ|p−1 and a(x, ξ) · ξ ≥ ν|ξ|p, x, ξ ∈ Rn, (2.5)

for some 0 < ν ≤ L, and V : Rn → [0,∞) does V ∈ Lγ0

loc(Rn). Note that the assumptions (1.3)
and (1.4) imply (2.5).

We recall the following two reverse Hölder type higher integrability results for the homogeneous
equation (2.4), see [24, Theorems 3.6 and 3.7] for their proofs.

Theorem 2.4. Assume a : Rn × Rn → Rn satisfies (2.5), and V : Rn → [0,∞) is such that
V ∈ Lγ0

loc(Rn) with γ0 given in (1.2) and V ∈ Bγ for some γ > 1. If h ∈ W 1,p(Ω2r) is a weak
solution to (2.4), then ( 1

|Br|

∫
Ωr

[V |h|s]γ dx
)1/γ

≤ c

|B2r|

∫
Ω2r

V |h|s dx

for some c = c(n, p, ν, L, s, γ, bγ) > 0.

Theorem 2.5. Assume a : Rn × Rn → Rn satisfies (1.3) and (1.4), and V : Rn → [0,∞) is
such that V ∈ Lγ0

loc(Rn) with γ0 given in (1.2) and V ∈ Bγ for some γ ∈ (p̃, n) with p̃ given
in (2.1). Then there exists a small δ = δ(n, p, L, ν, γ) > 0 such that the following holds: if a is
(δ,R)-vanishing, Ω is a (δ,R)-Reifenberg flat domain for some R > 0, and h ∈ W 1,p(Ω2r) is a
weak solution to (2.4), then(

−
∫
Ωr

|Dh|γ
∗(p−1) dx

) p
γ∗(p−1) ≤ c−

∫
Ω2r

|Dh|p dx

for some c = c(n, p, ν, L, s, γ, bγ) > 0.

Remark 2.6. In the above theorems, it may occur that γ < γ0. In this case, since V ∈ Lγ0 ,
we have V |h|s ∈ Lγ0(Ωr(x0)) in Theorem 2.4, and Dh ∈ Lγ∗

0 (p−1)(Ωr(x0),Rn) in Theorem 2.5.
However, the reverse Hölder-type estimates, where the constant c depends only on bγ for V , can
still be obtained with the exponent γ, owing to the assumption V ∈ Bγ .



EJDE-2025/102 DOUBLE OBSTACLE PROBLEMS 7

3. Comparison estimates

In this section, we derive comparison estimates with associated reference problems. Throughout
the following lemmas and proposition, let a : Rn × Rn → Rn satisfy (1.3) and (1.6) for some
L, ν̃ > 0, and V : Rn → [0,∞) belong to Lγ0

loc(Rn) with γ0 given in (1.2). Furthermore, assume
that F ∈ Lp(Ω,Rn) and ψ1, ψ2 ∈W 1,p(Ω) with ψ1 ≤ 0 ≤ ψ2 a.e. in Ω.

Before proceeding, we recall the following elementary inequalities, which will be used frequently
in the sequel. For every 1 < q <∞ and every ξ1, ξ2 ∈ Rd with d ∈ N, it holds

(|ξ1|2 + |ξ2|2)
q−2
2 |ξ2 − ξ1|2 ≤ c(q, d)

(
|ξ2|q−2ξ2 − |ξ1|q−2ξ1

)
· (ξ2 − ξ1) (3.1)

for some constant c(q, d) > 0. Moreover, if q ≥ 2,

|ξ2 − ξ1|q ≤ (|ξ1|2 + |ξ2|2)
q−2
2 |ξ2 − ξ1|2, (3.2)

and, if 1 < q < 2, we also have that for any κ ∈ (0, 1),

|ξ2 − ξ1|q = |ξ2 − ξ1|q
(
|ξ2|2 + |ξ1|2

) q(q−2)
4

(
|ξ2|2 + |ξ1|2

) q(2−q)
4

≤ κ
(
|ξ2|2 + |ξ1|2

)q/2
+ c(κ, q)

(
|ξ2|2 + |ξ1|2

) q−2
2 |ξ2 − ξ1|2

(3.3)

for some constant c(κ, q) > 0.
We start by recalling the following comparison lemma. For its proof, we refer, for instance, to

[6, Lemma 3.5].

Lemma 3.1. Let U be a bounded open set in Rn. Suppose that v1, v2 ∈ W 1,p(U) satisfy that

(v1 − v2)+ ∈W 1,p
0 (U) and∫

U

(a(x,Dv1)− a(x,Dv2)) ·D(v1 − v2)+ dx ≤ 0.

Then v1 ≤ v2 a.e. in U .

We derive comparison estimates. We first compare the function u ∈ A0(Ω), which satisfies
the variational inequality (1.7) corresponding to a double obstacle problem, with a function that
satisfies a variational inequality corresponding to a single obstacle problem in a local region. Fix
any x0 ∈ Ω and write Ωr = Ωr(x0).

Lemma 3.2. Let u ∈ A0(Ω) satisfy the variational inequality (1.7). If k ∈ Ãu(Ωr) := {f ∈
u+W 1,p

0 (Ωr) : f ≥ ψ1 a.e. in Ωr} satisfies the following variational inequality∫
Ωr

a(x,Dk) ·D(k − φ) dx+

∫
Ωr

V |k|s−2k(k − φ) dx ≤
∫
Ωr

a(x,Dψ2) ·D(k − φ) dx (3.4)

for all φ ∈ Ãu(Ωr). Then we have the following estimates:

(i) If p ≥ 2, then ∫
Ωr

|Du−Dk|p dx ≤ c

∫
Ωr

|F |p + |Dψ2|p dx (3.5)

and, for every ϵ ∈ (0, 1),∫
Ωr

V |u− k|s dx ≤ ϵ

∫
Ωr

V |u|s dx+ c(ϵ)

∫
Ωr

|F |p + |Dψ2|p dx. (3.6)

(ii) If 1 < p < 2, then for every ϵ ∈ (0, 1),∫
Ωr

|Du−Dk|p dx ≤ ϵ

∫
Ωr

|Du|p dx+ c(ϵ)

∫
Ωr

|F |p + |Dψ2|p dx (3.7)

and ∫
Ωr

V |u− k|s dx ≤ ϵ

∫
Ωr

|Du|p + V |u|s dx+ c(ϵ)

∫
Ωr

|F |p + |Dψ2|p dx. (3.8)

Here, c > 0 depends on n, p, L, ν̃, and s, while c(ϵ) > 0 depends on n, p, L, ν̃, s, and ϵ.



8 M. LEE, J. OK EJDE-2025/102

Proof. We note that φ := min{k, ψ2} = k− (k−ψ2)+ ∈ Ãu(Ωr). Plugging this φ into (3.4) yields∫
Ωr

(a(x,Dk)− a(x,Dψ2)) ·D(k − ψ2)+ dx ≤ −
∫
Ωr

V |k|p−2k(k − ψ2)+ dx ≤ 0,

where, in the last inequality, we have used the fact that k ≥ ψ2 ≥ 0 when (k − ψ2)+ ≥ 0. Hence,

by Lemma 3.1, k ≤ ψ2 a.e. in Ωr. Moreover, since k − u ∈W 1,p
0 (Ωr), we extend k to Ω\Ωr by u.

Then k ∈ A0(Ω), since u ∈ A0(Ω). Now we take the test functions φ := k and φ := u into (1.7)
and (3.4), respectively, and combine the two resulting estimates, to obtain∫

Ωr

(a(x,Du)− a(x,Dk)) ·D(u− k) dx+

∫
Ωr

(V |u|s−2u− V |k|s−2k)(u− k) dx

≤
∫
Ωr

(a(x,Dψ2)− |F |p−2F ) ·D(k − u) dx

≤ c

∫
Ωr

(|F |p−1 + |Dψ2|p−1)|Du−Dk| dx,

(3.9)

where we have used (1.3) in the last inequality.
We first consider the case p ≥ 2. Then we derive from the monotonicity condition (1.6) and

the elementary inequalities (3.1) and (3.2) that∫
Ωr

|Du−Dk|p dx+

∫
Ωr

V (|u|2 + |k|2)
s−2
2 |u− k|2 dx

≤ σ1

∫
Ωr

|Du−Dk|p dx+ c(σ1)

∫
Ωr

|F |p + |Dψ2|p dx

for any σ1 ∈ (0, 1), by using Young’s inequality. Taking σ1 = 1/2, we obtain∫
Ωr

|Du−Dk|p dx+

∫
Ωr

V (|u|2 + |k|2)
s−2
2 |u− k|2 dx ≤ c

∫
Ωr

|F |p + |Dψ2|p dx. (3.10)

Note that this directly implies (3.5) and, using (3.2), also implies (3.6) when s ≥ 2. We next prove
(3.6) when 1 < s < 2. By using (3.3) and (3.10), we derive that∫

Ωr

V |u− k|s dx ≤ σ2

∫
Ωr

V (|u|2 + |k|2) s
2 dx+ c(σ2)

∫
Ωr

V (|u|2 + |k|2)
s−2
2 |u− k|2 dx

≤ σ2

∫
Ωr

V (|u|s + |k|s) dx+ c(σ2)

∫
Ωr

|F |p + |Dψ2|p dx,
(3.11)

for any σ2 ∈ (0, 1). We first choose σ2 = 1/2, so that∫
Ωr

V |k|s dx ≤
∫
Ωr

V |u|s dx+

∫
Ωr

V |u− k|s dx

≤ 1

2

∫
Ωr

V |k|s dx+
3

2

∫
Ωr

V |u|s dx+ c

∫
Ωr

|F |p + |Dψ2|p dx,

which implies that ∫
Ωr

V |k|s dx ≤ c

∫
Ωr

V |u|s + |F |p + |Dψ2|p dx.

We insert this into (3.11), and obtain∫
Ωr

V |u− k|s dx ≤ cσ2

∫
Ωr

V |u|s dx+ c(σ2)

∫
Ωr

|F |p + |Dψ2|p dx,

which implies the estimate (3.6) when 1 < s < 2.
We next consider the case 1 < p < 2. From (3.9) with (1.6), (3.1), (3.3) and Young’s inequality,

we then derive that∫
Ωr

|Du−Dk|p dx+

∫
Ωr

V (|u|2 + |k|2)
s−2
2 |u− k|2 dx



EJDE-2025/102 DOUBLE OBSTACLE PROBLEMS 9

≤ σ3

∫
Ωr

(
|Du|2 + |Dk|2

) p
2 dx+ c(σ3)

∫
Ωr

(
|Du|2 + |Dk|2

) p−2
2 |Du−Dk|2 dx

+

∫
Ωr

V (|u|2 + |k|2)
s−2
2 |u− k|2 dx

≤ σ3

∫
Ωr

|Du|p + |Dk|p dx+ c(σ3)

∫
Ωr

(|F |p−1 + |Dψ2|p−1)|Du−Dk| dx

≤ σ3

∫
Ωr

|Du|p + |Dk|p dx+ σ1

∫
Ωr

|Du−Dk|p dx+ c(σ1, σ3)

∫
Ωr

|F |p + |Dψ2|p dx

for any σ3, σ1 ∈ (0, 1). By taking σ1 = 1/2, we have∫
Ωr

|Du−Dk|p dx+

∫
Ωr

V (|u|2 + |k|2)
s−2
2 |u− k|2 dx

≤ cσ3

∫
Ωr

|Du|p + |Dk|p dx+ c(σ3)

∫
Ωr

|F |p + |Dψ2|p dx.

We first select σ3 ∈ (0, 1) small, to obtain∫
Ωr

|Dk|p dx ≤ c

∫
Ωr

|Du|p + |F |p + |Dψ2|p dx.

Then inserting this into the previous estimate, we obtain∫
Ωr

|Du−Dk|p dx+

∫
Ωr

V (|u|2 + |k|2)
s−2
2 |u− k|2 dx

≤ cσ3

∫
Ωr

|Du|p dx+ c(σ3)

∫
Ωr

|F |p + |Dψ2|p dx
(3.12)

for any small σ3 ∈ (0, 1). This implies the estimate (3.7) and, using (3.2), also implies (3.8) when
s ≥ 2. It remains to prove (3.8) when 1 < s < 2. Using (3.3), we have from (3.12) that for any
σ4 ∈ (0, 1),∫

Ωr

V |u− k|s dx

≤ σ4

∫
Ωr

V (|u|2 + |k|2) s
2 dx+ c(σ4)

∫
Ωr

V (|u|2 + |k|2)
s−2
2 |u− k|2 dx

≤ cσ4

∫
Ωr

V (|u|s + |k|s) dx+ c(σ4)σ3

∫
Ωr

|Du|p dx+ c(σ3, σ4)

∫
Ωr

|F |p + |Dψ2|p dx.

(3.13)

We first select σ4 > 0 so small and σ3 = 1/2, to obtain∫
Ωr

V |k|s dx ≤ c

∫
Ωr

V |u|s + |Du|p + |F |p + |Dψ2|p dx.

Inserting this into (3.13), we obtain∫
Ωr

V |u− k|s dx

≤ cσ4

∫
Ωr

V |u|s dx+ σ3c(σ4)

∫
Ωr

|Du|p dx+ c(σ3, σ4)

∫
Ωr

|F |p + |Dψ2|p dx

for any small σ3, σ4 ∈ (0, 1). This implies the estimate (3.8) when 1 < s < 2. □

We next compare the function k in the above lemma with a weak solution to a homogeneous
quasilinear equation. Let v ∈W 1,p(Ωr) be a weak solution to

− div a(x,Dv) + V |v|s−2v = −div a(x,Dψ1) in Ωr,

v = k on ∂Ωr,
(3.14)



10 M. LEE, J. OK EJDE-2025/102

and h ∈W 1,p(Ωr) be a weak solution to

− div a(x,Dh) + V |h|s−2h = 0 in Ωr,

h = v on ∂Ωr.
(3.15)

Lemma 3.3. Let k ∈ Ãu(Ωr) satisfy the variational inequality (3.4) given in Lemma 3.2, and
h ∈W 1,p(Ωr) be a weak solution to (3.15). Then we have the following estimates:

(i) If p ≥ 2, ∫
Ωr

|Dk −Dh|p dx ≤ c

∫
Ωr

|Dψ1|p + |Dψ2|p dx, (3.16)

and, for every ϵ ∈ (0, 1),∫
Ωr

V |k − h|s dx ≤ ϵ

∫
Ωr

V |k|s dx+ c(ϵ)

∫
Ωr

|Dψ1|p + |Dψ2|p dx. (3.17)

(ii) If 1 < p < 2, for every ϵ ∈ (0, 1),∫
Ωr

|Dk −Dh|p dx ≤ ϵ

∫
Ωr

|Dk|p dx+ c(ϵ)

∫
Ωr

|Dψ1|p + |Dψ2|p dx, (3.18)

and ∫
Ωr

V |k − h|s dx ≤ ϵ

∫
Ωr

|Dk|p + V |k|s dx+ c(ϵ)

∫
Ωr

|Dψ1|p + |Dψ2|p dx. (3.19)

Here, c > 0 depends on n, p, L, ν̃, and s, while c(ϵ) > 0 depends on n, p, L, ν̃, s, and ϵ.

Proof. Since v = k = u on ∂Ωr in the trace sense, we have (ψ1 − v)+ ∈ W 1,p
0 (Ωr). Then by

adopting the test function (ψ1 − v)+ in the weak formulation of (3.14), we have∫
Ωr

(a(x,Dψ1)− a(x,Dv)) ·D(ψ1 − v)+ dx =

∫
Ωr

V |v|s−2v(ψ1 − v)+ dx ≤ 0,

where, in the last inequality, we have used the fact that v ≤ ψ1 ≤ 0 when (ψ1 − v)+ ≥ 0. Then by

Lemma 3.1 with v1 := ψ1 and v2 := v, we have v ≥ ψ1 a.e. in Ωr. Therefore, v ∈ Ãu(Ωr) and so
we take v as a test function into (3.4) in order to infer that∫

Ωr

a(x,Dk) ·D(k − v) dx+

∫
Ωr

V |k|s−2k(k − v) dx

≤
∫
Ωr

a(x,Dψ2) ·D(k − v) dx.

Furthermore, we take v − k ∈ W 1,p
0 (Ωr) as a test function in the weak formulation of (3.14) to

discover that ∫
Ωr

a(x,Dv) ·D(v − k) dx+

∫
Ωr

V |v|s−2v(v − k) dx

=

∫
Ωr

a(x,Dψ1) ·D(v − k) dx.

Combining the above two estimates and using (1.3), we obtain∫
Ωr

(a(x,Dk)− a(x,Dv)) ·D(k − v) dx+

∫
Ωr

(V |k|s−2k − V |v|s−2v)(k − v) dx

≤ c

∫
Ωr

(|Dψ1|p−1 + |Dψ2|p−1)|Dk −Dv| dx.

We notice that this estimate is the same as (3.9) with u, k, and |F |p−1 + |Dψ2|p−1 replaced by
k, v, and |Dψ1|p−1 + |Dψ2|p−1, respectively. Therefore, by repeating the same arguments used in
the proof of Lemma 3.2 for (3.9), we obtain the following estimates: for ϵ ∈ (0, 1): if p ≥ 2, then∫

Ωr

|Dk −Dv|p dx ≤ c

∫
Ωr

|Dψ1|p + |Dψ2|p dx (3.20)



EJDE-2025/102 DOUBLE OBSTACLE PROBLEMS 11

and ∫
Ωr

V |k − v|s dx ≤ ϵ

∫
Ωr

V |k|s dx+ c(ϵ)

∫
Ωr

|Dψ1|p + |Dψ2|p dx; (3.21)

and if 1 < p < 2, then∫
Ωr

|Dk −Dv|p dx ≤ ϵ

∫
Ωr

|Dk|p dx+ c(ϵ)

∫
Ωr

|Dψ1|p + |Dψ2|p dx (3.22)

and ∫
Ωr

V |k − v|s dx ≤ ϵ

∫
Ωr

|Dk|p + V |k|s dx+ c(ϵ)

∫
Ωr

|Dψ1|p + |Dψ2|p dx. (3.23)

We next test the equations (3.14) and (3.15) with the test function φ := v − h ∈ W 1,p
0 (Ωr) to

derive ∫
Ωr

a(x,Dv) ·D(v − h) dx+

∫
Ωr

V |v|s−2v(v − h) dx =

∫
Ωr

a(x,Dψ1) ·D(v − h) dx

and ∫
Ωr

a(x,Dh) ·D(v − h) dx+

∫
Ωr

V |h|s−2h(v − h) dx = 0.

Combining these estimates and using (1.3), we have∫
Ωr

(a(x,Dv)− a(x,Dh)) ·D(v − h) dx+

∫
Ωr

(V |v|s−2v − V |h|s−2h)(v − h) dx

≤ c

∫
Ωr

|Dψ1|p−1|Dv −Dh| dx.

We observe that this estimate has the same structure as (3.9) with u, k, and |F |p−1 + |Dψ2|p−1

replaced by v, h, and |Dψ1|p−1, respectively. Therefore, by applying the same argument used for
(3.9) in the proof of Lemma 3.2, we obtain the following estimate: for ϵ ∈ (0, 1), if p ≥ 2, then∫

Ωr

|Dv −Dh|p dx ≤ c

∫
Ωr

|Dψ1|p dx (3.24)

and ∫
Ωr

V |v − h|s dx ≤ ϵ

∫
Ωr

V |v|s dx+ c(ϵ)

∫
Ωr

|Dψ1|p dx

≤ cϵ
(∫

Ωr

V |k − v|s dx+

∫
Ωr

V |k|s dx
)
+ c(ϵ)

∫
Ωr

|Dψ1|p dx;
(3.25)

and if 1 < p < 2, then∫
Ωr

|Dv −Dh|p dx ≤ ϵ

∫
Ωr

|Dv|p dx+ c(ϵ)

∫
Ωr

|Dψ1|p dx

≤ cϵ
(∫

Ωr

|Dk −Dv|p dx+

∫
Ωr

|Dk|p dx
)
+ c(ϵ)

∫
Ωr

|Dψ1|p dx,
(3.26)

and ∫
Ωr

V |v − h|s dx ≤ ϵ

∫
Ωr

|Dv|p + V |v|s dx+ c(ϵ)

∫
Ωr

|Dψ1|p dx

≤ cϵ

∫
Ωr

|Dk −Dv|p + V |k − v|s dx

+ cϵ

∫
Ωr

|Dk|p + V |k|s dx+ c(ϵ)

∫
Ωr

|Dψ1|p dx.

(3.27)

Therefore, combining (3.20)–(3.23) and (3.24)–(3.27), we arrive at the desired estimates. □

Combining Lemmas 3.2 and 3.3, we obtain the following comparison estimates.

Proposition 3.4. Let u ∈ A0(Ω) satisfy the variational inequality (1.7), and h ∈ W 1,p(Ωr) be
the weak solution to (3.15). Then we have the estimates:



12 M. LEE, J. OK EJDE-2025/102

(i) (Energy estimate)∫
Ωr

|Dh|p dx ≤ c

∫
Ωr

|Du|p + |F |p + |Dψ1|p + |Dψ2|p dx, (3.28)

and ∫
Ωr

V |h|s dx ≤ c

∫
Ωr

V |u|s + |F |p + |Dψ1|p + |Dψ2|p + |Du|pχ{p<2} dx, (3.29)

where χ{p<2} = 1 if 1 < p < 2 and χ{p<2} = 0 if p ≥ 2.
(ii) If p ≥ 2, ∫

Ωr

|Du−Dh|p dx ≤ c

∫
Ωr

|F |p + |Dψ1|p + |Dψ2|p dx, (3.30)

and, for every ϵ ∈ (0, 1),∫
Ωr

V |u− h|s dx ≤ ϵ

∫
Ωr

V |u|s dx+ c(ϵ)

∫
Ωr

|F |p + |Dψ1|p + |Dψ2|p dx. (3.31)

(iii) If 1 < p < 2, for every ϵ ∈ (0, 1),∫
Ωr

|Du−Dh|p dx ≤ ϵ

∫
Ωr

|Du|p dx+ c(ϵ)

∫
Ωr

|F |p + |Dψ1|p + |Dψ2|p dx, (3.32)

and∫
Ωr

V |u− h|s dx ≤ ϵ

∫
Ωr

|Du|p + V |u|s dx+ c(ϵ)

∫
Ωr

|F |p + |Dψ1|p + |Dψ2|p dx. (3.33)

Here, c > 0 depends on n, p, L, ν̃, and s, while c(ϵ) > 0 depends on n, p, L, ν̃, s, and ϵ.

Proof. We first observe from (3.5) and (3.16), when p ≥ 2, and from (3.7) and (3.18), when
1 < p < 2, that ∫

Ωr

|Dk|p dx ≤ c

∫
Ωr

|Du|p + |F |p + |Dψ2|p dx, (3.34)

and hence ∫
Ωr

|Dh|p dx ≤ c

∫
Ωr

|Dk|p + |Dψ1|p + |Dψ2|p dx

≤ c

∫
Ωr

|Du|p + |F |p + |Dψ1|p + |Dψ2|p dx.

Note that the last inequality verifies (3.28). Moreover, from (3.6) and (3.17), when p ≥ 2, and
from (3.8) and (3.19), when 1 < p < 2, that∫

Ωr

V |k|s dx ≤ c

∫
Ωr

V |u|s + |F |p + |Dψ2|p + |Du|pχ{p<2} dx, (3.35)

and hence ∫
Ωr

V |h|s dx ≤ c

∫
Ωr

V |k|s + |Dψ1|p + |Dψ2|p + |Dk|pχ{p<2} dx

≤ c

∫
Ωr

V |u|s + |F |p + |Dψ1|p + |Dψ2|p + |Du|pχ{p<2} dx.

Note that the last inequality verifies (3.29).
Finally, by plugging (3.34) and (3.35) into the estimates in Lemma 3.3 and combining these

with the estimates in Lemma 3.2, we obtain the desired estimates (3.30)-(3.33). □

Remark 3.5. By carefully following the proofs of the above lemmas in the case s ≥ 2, we can
actually obtain the following sharper comparison estimates:∫

Ωr

V |u− h|s dx ≤ c

∫
Ωr

|F |p + |Dψ1|p + |Dψ2|p dx,



EJDE-2025/102 DOUBLE OBSTACLE PROBLEMS 13

when p ≥ 2, and∫
Ωr

V |u− h|s dx ≤ ϵ

∫
Ωr

|Du|p dx+ c(ϵ)

∫
Ωr

|F |p + |Dψ1|p + |Dψ2|p dx,

for every ϵ ∈ (0, 1), when 1 < p < 2. Note that these bounds are sharper than the estimates
(3.31) and (3.33), respectively. However, they do not yield any additional advantage in the proof
of Theorem 1.1 in the next section. For this reason, we proceed with the estimates (3.31) and
(3.33), which are valid for both the cases 1 < s < 2 and 2 ≤ s < p∗.

4. Lq estimates

We are now ready to prove our main results.

Proof of Theorem 1.1. We follow the approach introduced by Mingione in [1, 30]. The overall
proof strategy is essentially the same as in [24, Theorem 2.3], so we omit some of the technical
details.

Step 1. (Setting) Assume a : Rn × Rn → Rn satisfies (1.3) and (1.4) and V : Rn → [0,∞) is
such that V ∈ Lγ0

loc(Rn) with γ0 given in (1.2). Furthermore, we assume that a : Rn × Rn → Rn

is (δ,R)-vanishing and Ω is (δ,R)-Reifenberg flat for some R > 0, where δ ∈ (0, 1) will be chosen
sufficiently small later in Step 3 (see Remark 4.2), and that V ∈ Bγ , where the range of γ is given
in (1)–(3) of Theorem 1.1. Finally, let F ∈ Lp(Ω,Rn) and ψ1, ψ2 ∈ W 1,p(Ω) with ψ1 ≤ 0 ≤ ψ2

a.e. in Ω. Fix any x0 ∈ Ω and r > 0 satisfying r ≤ R/2.
We prove the estimates in (1)–(3) of Theorem 1.1, specifically (1.9)-(1.11), at one time by

defining the function Φ(w;x) for w ∈ W 1,p(Ω) and the constant γ1 differently in each case as
follows:

Case 1: Estimation of (1.9). Let 1 < p < ∞ and p̃ < γ < ∞ with p̃ given in (2.1). We fix

q ∈ (1, γ
∗(p−1)

p ) with γ∗ given in (2.1), and denote

Φ(w) = Φ(w;x) := |Dw(x)|p and γ1 :=

{
γ∗(p−1)

p if γ < n,

q + 1 if γ ≥ n.

Note that, when γ ≥ n, there exists γ2 ∈ (1, n) such that q + 1 =
γ∗
2 (p−1)

p since (q+1)p
p−1 > 2.

Case 2: Estimation of (1.10). Let p ≥ 2 and 1 < γ <∞. We fix any q ∈ (1, γ), and denote

Φ(w) = Φ(w;x) := V (x)|w(x)|s and γ1 := γ.

Case 3: Estimation of (1.11). Let 1 < p < 2 and n/p ≤ γ < ∞. We fix any q ∈ (1, γ), and
denote

Φ(w) = Φ(w;x) := |Dw(x)|p + V (x)|w(x)|s and γ1 := γ.

Since γ1 ≤ γ, we note that V ∈ Bγ1 in all the above cases. With the function Φ defined above
and u ∈ A0(Ω) satisfying the variational inequality (1.7), we define

E(λ, ρ) := {x ∈ Ωρ : Φ(u;x) > λ} for λ > 0,

and

λ0 := −
∫
Ω2r

Φ(u) dx+
1

δ1
−
∫
Ω2r

Ψ dx, (4.1)

where δ1 ∈ (0, 1) will be chosen sufficiently small later in Step 4, and Ψ is given in (1.8). Finally,
fix any τ1, τ2 such that 1 ≤ τ1 < τ2 ≤ 2, and then it follows that Ωr ⊂ Ωτ1r ⊂ Ωτ2r ⊂ Ω2r.

Step 2. (Covering lemma) We consider λ > 0 large enough such that

λ > αλ0, where α :=
(16
7

)n( 20

τ2 − τ1

)n

. (4.2)

Note that ( 167 )n is originated from (2.3). Then we have the following lemma. For the proof we
refer to [24, Lemma 4.3].



14 M. LEE, J. OK EJDE-2025/102

Lemma 4.1. Given λ > αλ0, there exists a disjoint family {Ωρi
(yi)}∞i=1 with yi ∈ E(λ, τ1r) and

ρi ∈
(
0, (τ2−τ1) r

10

)
such that

E(λ, τ1r) \ N ⊂
∞⋃
i=1

Ω5ρi
(yi)

for some measure zero set N ,

−
∫
Ωρi

(yi)

Φ(u) dx+
1

δ1
−
∫
Ωρi

(yi)

Ψ dx = λ, (4.3)

and for any ρ ∈ (ρi, (τ2 − τ1) r],

−
∫
Ωρ(yi)

Φ(u) dx+
1

δ1
−
∫
Ωρ(yi)

Ψ dx < λ. (4.4)

Furthermore, from (4.3) we have

|Ωρi
(yi)| ≤ 2

λ

(∫
Ωρi

(yi)∩{Φ(u)>λ
4 }

Φ(u) dx+
1

δ1

∫
Ωρi

(yi)∩{Ψ>
δ1λ
4 }

Ψ dx
)
. (4.5)

Step 3. (Comparison estimates) For each i, we note from (4.4) in Lemma 4.1 that

−
∫
Ω10ρi

(yi)

Φ(u) dx+
1

δ1
−
∫
Ω10ρi

(yi)

Ψ dx < λ.

Applying Proposition 3.4 (ii) and (iii) with r = 10ρi, we have that for any ϵ ∈ (0, 1), there exists
a small δ1 = δ1(ϵ, n, p, L, ν, s) ∈ (0, 1) such that

−
∫
Ω10ρi

(yi)

Φ(u− hi) dx ≤ ϵ−
∫
Ω10ρi

(yi)

Φ(u) dx+ c(ϵ)−
∫
Ω10ρi

(yi)

Ψ dx

≤ ϵλ+ c(ϵ)δ1λ ≤ 2ϵλ

(4.6)

(In fact, when p ≥ 2 in Case 1, the constant c(ϵ) in (4.6) can be chosen as a constant independent
of ϵ), where hi ∈W 1,p(Ω10ρi(y

i)) is a weak solution to

−div a(x,Dhi) + V |hi|s−2hi = 0 in Ω10ρi
(yi),

hi = 0 on ∂wΩ10ρi
(yi) if B10ρi

(x0) ̸⊂ Ω.

Furthermore, applying Theorem 2.4 to Case 2 and Case 3, and Theorem 2.5 to Case 1 and Case
3 with γ = γ1, where γ1 defined in each case, and also applying Proposition 3.4 (i), we have(

−
∫
Ω5ρi

(yi)

Φ(hi)
γ1 dx

)1/γ1

≤ c−
∫
Ω10ρi

(yi)

Φ(hi) dx ≤ c−
∫
Ω10ρi

(yi)

Φ(u) + Ψ dx ≤ cλ, (4.7)

for some c = c(n, p, L, ν, s, γ, bγ) > 0.

Remark 4.2. At this stage, the constant δ is fixed as the one in Theorem 2.5 with γ = γ1.
Therefore, δ depends on n, p, L, ν, γ when γ < n or n, p, L, ν, q when γ ≥ n. Moreover, in Case 2,
Theorem 2.5 is not used; instead, only Theorem 2.4 is applied. Hence, the (δ,R)-vanishing and
(δ,R)-Reifenberg flat assumptions on a and Ω, respectively, are not needed to obtain (1.10).

Remark 4.3. In Case 3, we assume γ1 = γ ≥ n
p , which is equivalent to γ1 ≤ γ∗1

p−1
p . Therefore,

we can apply both Theorem 2.5 and Theorem 2.4 with exponent γ1 in this case.

Let x ∈ Ω5ρi
(yi) such that Φ(u;x) > Kλ, where the constant K > 1 will be chosen sufficiently

large later in Step 4. For this x, it follows from simple calculation that

Φ(u;x) ≤ 2max{p,s}−1
(
Φ(u− hi;x) + Φ(hi;x)

)
≤ 2max{p,s}Φ(u− hi;x) +

2γ1 max{p,s}

(Kλ)γ1−1
Φ(hi;x)

γ1 .



EJDE-2025/102 DOUBLE OBSTACLE PROBLEMS 15

Then we apply (4.6), (4.7), and (2.3) to derive∫
Ω5ρi

(yi)∩E(Kλ,τ2r)

Φ(u) dx ≤ c

∫
Ω5ρi

(yi)

Φ(u− hi) dx+
c

(Kλ)γ1−1

∫
Ω5ρi

(yi)

Φ(hi)
γ1 dx

≤ c
(
ϵλ+

λγ1

(Kλ)γ1−1

)
|Ω5ρi

(yi)|

≤ cλ
(
ϵ+K1−γ1

) ∣∣Ωρi(y
i)
∣∣

= cϵ̃λ
∣∣Ωρi(y

i)
∣∣

for a constant c = c(n, p, L, ν, s, γ, bγ) > 0, where

ϵ̃ := ϵ+K1−γ1 . (4.8)

By inserting (4.5) into the preceding estimate, we obtain that∫
Ω5ρi

(yi)∩E(Kλ,τ2r)

Φ(u) dx ≤ cϵ̃
(∫

Ωρi
(yi)∩{Φ(u)>λ

4 }
Φ(u) dx+

1

δ1

∫
Ωρi

(yi)∩{Ψ>
δ1λ
4 }

Ψ dx
)
.

From Lemma 4.1, we observe that the sets Ωρi(y
i) are pairwise disjoint and the following inclusion

relations hold

E(Kλ, τ1r) \ N ⊂ E(λ, τ1r) \ N ⊂ ∪∞
i=1Ω5ρi(y

i) ⊂ Ωτ2r,

since K > 1. Therefore,∫
E(Kλ,τ1r)

Φ(u) dx ≤
∞∑
i=1

∫
Ω5ρi

(yi)∩E(Kλ,τ1r)

Φ(u) dx

≤ cϵ̃
(∫

Ωτ2r∩{Φ(u)>λ
4 }

Φ(u) dx+
1

δ1

∫
Ωτ2r∩{Ψ>

δ1λ
4 }

Ψ dx
) (4.9)

for a constant c = c(n, p, L, ν, s, γ, bγ) > 0.

Step 4. (Proof of (1.9)–(1.11)) It remains to conclude the proof via a truncation argument
employing Fubini’s theorem. Suppose that∫

Ω2r

Ψq dx <∞.

For k > 0, let us define

Φk(u) = Φk(u;x) := min {Φ(u;x), k} ,
and consider the super-level set with respect to Φk(u) as

Ek(λ̃, ρ) := {x ∈ Ωρ : Φk(u;x) > λ̃} for λ̃, ρ > 0.

Then, since Ek(λ̃, ρ) = ∅ when k ≤ λ̃ and Ek(λ̃, ρ) = E(λ̃, ρ) when k > λ̃, it follows from (4.9)
that ∫

Ek(Kλ,τ1r)

Φ(u) dx ≤ cϵ̃
(∫

Ek(λ
4 ,τ2r)

Φ(u) dx+
1

δ1

∫
Ωτ2r∩{Ψ>

δ1λ
4 }

Ψ dx
)
.

Multiplying both sides by λq−2, integrating with respect to λ over (αλ0,∞), and using Fubini’s
theorem, we derive that∫

Ek(Kαλ0,τ1r)

Φ(u)
[ ∫ Φk(u)/K

αλ0

λq−2 dλ
]
dx

=

∫ ∞

αλ0

λq−2

∫
Ek(Kλ,τ1r)

Φ(u) dx dλ

≤ cϵ̃
(∫ ∞

αλ0

λq−2

∫
Ek(λ

4 ,τ2r)
Φ(u) dxdλ+

∫ ∞

αλ0

λq−2

∫
Ωτ2r∩

{
Ψ
δ1

>λ
4

} Ψ

δ1
dxdλ

)
≤ cϵ̃

(∫
Ek(αλ0

4 ,τ2r)
Φ(u)

[ ∫ 4Φk(u)

αλ0

λq−2 dλ
]
dx



16 M. LEE, J. OK EJDE-2025/102

+

∫
Ωτ2r∩

{
Ψ
δ1

>
αλ0
4

} Ψ

δ1

[ ∫ 4Ψ/δ1

αλ0

λq−2 dλ
]
dx

)
≤ cϵ̃

(∫
Ωτ2r

Φ(u)Φk(u)
q−1 dx+

∫
Ωτ2r

[Ψ
δ1

]q
dx

)
.

Then ∫
Ek(Kαλ0,τ1r)

Φ(u)Φk(u)
q−1 dx

= (q − 1)Kq−1
{
(αλ0)

q−1

∫
Ek(Kαλ0,τ1r)

Φ(u) dx+
(αλ0)

q−1

q − 1

∫
Ek(Kαλ0,τ1r)

Φ(u) dx
}

≤ (Kαλ0)
q−1

∫
Ωτ1r

Φ(u) dx+ cϵ̃Kq−1
(∫

Ωτ2r

Φ(u)Φk(u)
q−1 dx+

∫
Ωτ2r

[Ψ
δ1

]q
dx

)
.

It can also be seen that∫
Ωτ1r\Ek(Kαλ0,τ1r)

Φ(u)Φk(u)
q−1 dx ≤ (Kαλ0)

q−1

∫
Ωτ1r

Φ(u) dx.

As a result of the previous two estimates, we obtain∫
Ωτ1r

Φ(u)Φk(u)
q−1 dx

≤ c(Kαλ0)
q−1

∫
Ωτ1r

Φ(u) dx+ c2ϵ̃K
q−1

(∫
Ωτ2r

Φ(u)Φk(u)
q−1 dx+

∫
Ωτ2r

[Ψ
δ1

]q
dx

)
for a constant c2 = c2(n, p, L, ν, s, γ, bγ , q) > 0. At this stage, we recall the definition of ϵ̃ given in
(4.8), and then choose a sufficiently large K > 1 and a sufficiently small ϵ ∈ (0, 1) depending on
n, p, L, ν, s, γ, bγ , q such that

K ≥ (4c2)
1

γ1−q and ϵ ≤ 1

4c2Kq−1
,

which implies

c2ϵ̃K
q−1 = c2(ϵK

q−1 +Kq−γ1) ≤ 1

2
,

hence δ1 = δ1(n, p, L, ν, s, γ, bγ , q) ∈ (0, 1) is definitively determined. Recalling the definition of α
in (4.2), we thus conclude that∫

Ωτ1r

Φ(u)Φk(u)
q−1 dx

≤ 1

2

∫
Ωτ2r

Φ(u)Φk(u)
q−1 dx+

cλq−1
0

(τ2 − τ1)n

∫
Ω2r

Φ(u) dx+ c

∫
Ω2r

Ψq dx.

Applying Lemma 2.1, we obtain∫
Ωr

Φ(u)Φk(u)
q−1 dx ≤ cλq−1

0

∫
Ω2r

Φ(u) dx+ c

∫
Ω2r

Ψq dx

for any large k > 0. Finally, invoking Lebesgue’s monotone convergence theorem along with
Hölder’s inequality, Young’s inequality, and the definition of λ0 given in (4.1), we conclude that

−
∫
Ωr

Φ(u)q dx = lim
k→∞

−
∫
Ωr

Φ(u)Φk(u)
q−1 dx ≤ cλq−1

0 −
∫
Ω2r

Φ(u) dx+ c−
∫
Ω2r

Ψq dx

≤ c
(
−
∫
Ω2r

Φ(u) dx
)q

+ c−
∫
Ω2r

Ψq dx.

This yields the desired estimates (1.9)–(1.11), upon recalling the definition of Φ(u) in Cases 1–3.



EJDE-2025/102 DOUBLE OBSTACLE PROBLEMS 17

Acknowledgements. M. Lee was supported by the National Research Foundation of Korea(NRF)
grant funded by the Korea government(MSIT)(RS-2025-23525636). J. Ok was supported by the
National Research Foundation of Korea funded by the Korean Government (NRF-2022R1C1C1004523).

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Mikyoung Lee
Department of Mathematics and Institute of Mathematical Science, Pusan National University, Busan
46241, Korea

Email address: mikyounglee@pusan.ac.kr

Jihoon Ok

Department of Mathematics, Institute for Mathematical and Data Science, Sogang University, Seoul
04107, Korea

Email address: jihoonok@sogang.ac.kr